<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">D</font> being  <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a>  holds  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Olive" title="b1">D</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">D</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c2">D</font></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c3">A</font> = <font color="Maroon" title="c1">C</font> as   <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#D2" title="JORDAN2C:def.2">JORDAN2C:def 2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>, <font color="Maroon" title="c5">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E3:17"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r1</font>
 <span class="kw">and </span><br/><a NAME="E4:17"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c4">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_6.html#D10" title="METRIC_6:def.10">METRIC_6:def 10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">B</font> = <font color="Maroon" title="c2">D</font> as   <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#D2" title="JORDAN2C:def.2">JORDAN2C:def 2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">r2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>, <font color="Maroon" title="c8">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E7:17"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r2</font>
 <span class="kw">and </span><br/><a NAME="E8:17"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c7">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_6.html#D10" title="METRIC_6:def.10">METRIC_6:def 10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">C9</font> = <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c4">r1</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> , <font color="Maroon" title="c10">D9</font> = <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c7">r2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1k.html#T37" title="JORDAN1K:th.37">JORDAN1K:37</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>, <font color="Maroon" title="c12">r3</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E11:17"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c4">r1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c7">r2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">x3</font>,<font color="Maroon" title="c12">r3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="weierstr.html#T1" title="WEIERSTR:th.1">WEIERSTR:1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E17">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:17_1"/>
<font color="Maroon" title="c10">D9</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:17_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">D9</font> is   <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#D2" title="JORDAN2C:def.2">JORDAN2C:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:17_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1k.html#T8" title="JORDAN1K:th.8">JORDAN1K:8</a>, <a class="ref" href="tbsp_1.html#T19" title="TBSP_1:th.19">TBSP_1:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E18">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:17_2"/>
<font color="Maroon" title="c9">C9</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:17_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">C9</font> is   <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#D2" title="JORDAN2C:def.2">JORDAN2C:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:17_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1k.html#T8" title="JORDAN1K:th.8">JORDAN1K:8</a>, <a class="ref" href="tbsp_1.html#T19" title="TBSP_1:th.19">TBSP_1:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E14:17"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">x3</font>,<font color="Maroon" title="c12">r3</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c4">r1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c7">r2</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:17"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="subset_1.html#T31" title="SUBSET_1:th.31">SUBSET_1:31</a></span>;<br/>
<a NAME="E15:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A6</font></span>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">x3</font>,<font color="Maroon" title="c12">r3</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c4">r1</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c7">r2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T53" title="XBOOLE_1:th.53">XBOOLE_1:53</a></span>;<br/>
<a NAME="E16:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A7</font></span>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">x3</font>,<font color="Maroon" title="c12">r3</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c4">r1</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T18" title="XBOOLE_1:th.18">XBOOLE_1:18</a></span>;<br/>
<a NAME="E17:17"/><span class="lab"><font color="Green" title="E21">A8</font></span>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">x3</font>,<font color="Maroon" title="c12">r3</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c7">r2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:17"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="ref" href="xboole_1.html#T18" title="XBOOLE_1:th.18">XBOOLE_1:18</a></span>;<br/>
<a NAME="E18:17"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c7">r2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:17"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="subset_1.html#T31" title="SUBSET_1:th.31">SUBSET_1:31</a></span>;<br/>
<a NAME="E19:17"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c7">r2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T43" title="SUBSET_1:th.43">SUBSET_1:43</a></span>;<br/>
<a NAME="E20:17"/><span class="kw">then </span>
<font color="Maroon" title="c10">D9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:17"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="jordan2c.html#T133" title="JORDAN2C:th.133">JORDAN2C:133</a></span>;<br/>
<a NAME="E21:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A9</font></span>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">x3</font>,<font color="Maroon" title="c12">r3</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:17"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E22:17"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c4">r1</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="subset_1.html#T31" title="SUBSET_1:th.31">SUBSET_1:31</a></span>;<br/>
<a NAME="E23:17"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c4">r1</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T43" title="SUBSET_1:th.43">SUBSET_1:43</a></span>;<br/>
<a NAME="E24:17"/><span class="kw">then </span>
<font color="Maroon" title="c9">C9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:17"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="ref" href="jordan2c.html#T133" title="JORDAN2C:th.133">JORDAN2C:133</a></span>;<br/>
<a NAME="E25:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A10</font></span>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">x3</font>,<font color="Maroon" title="c12">r3</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:17"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E26:17"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">x3</font>,<font color="Maroon" title="c12">r3</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1k.html#T8" title="JORDAN1K:th.8">JORDAN1K:8</a>, <a class="ref" href="tbsp_1.html#T19" title="TBSP_1:th.19">TBSP_1:19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E27:17"/>
 <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c2">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:17"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="txt" href="#E21:17"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="ref" href="xboole_1.html#T68" title="XBOOLE_1:th.68">XBOOLE_1:68</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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